Question

Aratati ca sin10*cos20*cos40=1/8

Answer 1

Stim ca avem urmatoarea formula
$\sin{2x}=2\sin{x}\cos{x}$
Inmultim ecuatia de mai sus cu cos10
$\sin{10}*\cos{10}*cos{20}*\cos{40}=\frac{\cos{10}}{2*2*2}\Rightarrow 2\sin{10}*\cos{10}*\cos{20}*\cos{40}=\sin{20}\cos{20}\cos{40}=\frac{\cos{10}}{2*2}\Rightarrow 2\sin{20}\cos{20}\cos{40}=\sin{40}\cos{40}=\frac{\cos{10}}{2}\Rightarrow 2\sin{40}\cos{40}=\sin{80}=\cos{10}$
Dar stim ca in general
$\sin{\frac{\pi}{2}-x}=\sin{(90-x)}=\cos{x}$
Daca inlocuim pe x cu 10 obtinem
$\sin{80}=\sin{(90-10)}=\cos{10}$ adica exact relatia ce trebuia demonstrata